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We report the results on the proton mass decomposition and also on the related quark and glue momentum fractions. The results are based on overlap valence fermions on four ensembles of N f = 2 + 1 DWF configurations with three lattice spacings and volumes, and several pion masses including the physical pion mass. With 1-loop pertur-bative calculation and proper normalization of the glue operator, we find that the u , d , and s quark masses contribute 9(2)% to the proton mass. The quark energy and glue field energy contribute 31(5)% and 37(5)% respectively in the MS scheme at µ = 2 GeV. The trace anomaly gives the remaining 23(1)% contribution. The u , d , s and glue momentum fractions in the MS scheme are consistent with the global analysis at µ = 2 GeV.

We present a lattice calculation of the leading order (LO) hadronic vacuum polarization (HVP) contribution to the muon anomalous magnetic moment for the connected light and strange quarks, \\(a^{\\rm W}_{{\\rm con}, l/s}\\) in the widely used window \\(t_0=0.4~\\mathrm{fm}\\), \\(t_1=1.0~\\mathrm{fm}\\), \\(\\Delta=0.15~\\mathrm{fm}\\), and also of \\(a^{\\rm S}_{{\\rm con}, l/s}\\) in the short distance region. We use overlap fermions on 4 physical-point ensembles. Two 2+1 flavor RBC/UKQCD ensembles use domain wall fermions (DWF) and Iwasaki gauge actions at \\(a = 0.084\\) and 0.114 fm, and two 2+1+1 flavor MILC ensembles use the highly improved staggered quark (HISQ) and Symanzik gauge actions at \\(a = 0.088\\) and 0.121 fm. We have incorporated infinite volume corrections from 3 additional DWF ensembles at \\({\\rm L}\\) = 4.8, 6.4 and 9.6 fm and physical pion mass. For \\(a^{\\rm W}_{{\\rm con}, l}\\), we find that our results on the two smaller lattice spacings are consistent with those using the unitary setup, but those at the two coarser lattice spacings are slightly different. Eventually, we predict \\(a^{\\rm W}_{{\\rm con}, l}=206.7(1.5)(1.0)\\) and \\(a^{\\rm W}_{{\\rm con}, s}=26.8(0.1)(0.3)\\), using linear extrapolation in \\(a^2\\), with systematic uncertainties estimated from the difference of the central values from the RBC/UKQCD and MILC ensembles.

The hadronic tensor is the central non-perturbative object in the calculation of the cross section of lepton-hadron interactions like neutrino-nucleon scattering. It is usually parameterized in terms of structure functions, which encode all necessary information for all kinematic regions. Moreover, the structure functions can be factorized in terms of parton distribution functions (PDFs) and contains information on hadron resonances. On the lattice, we can calculate the corresponding matrix element of two quark-bilinear currents with a relative Euclidean time separation. The reconstruction of the hadronic tensor in Minkowski space requires appropriate dealing with the corresponding inverse problem. In our current work, we extend previous calculations on the nucleon by considering a much larger range of momentum transfers, which is inevitable in the context of structure functions. This can be achieved by using stochastic sources, which allows us to calculate the required four-point functions in a broad kinematic region. We employ a clover fermion ensemble at pion mass \\(m_\\pi = 223~\\mathrm{MeV}\\) and lattice spacing \\(a=0.085~\\mathrm{fm}\\). In these proceedings, we will give an overview of our simulation and present some first preliminary results.

We present a calculation of the proton momentum and angular momentum decompositions using overlap fermions on a \\(2+1\\)-flavor RBC/UKQCD domain-wall lattice at 0.143 fm with a pion mass of 171 MeV which is close to the physical one. A complete determination of the momentum and angular momentum fractions carried by up, down, strange and glue inside the proton has been done with valence pion masses varying from 171 to 391 MeV. We have utilized fast Fourier transform on the stochastic-sandwich method for connected-insertion parts and the cluster-decomposition error reduction technique for disconnected-insertion parts has been used to reduce statistical errors. The full nonperturbative renormalization and mixing between the quark and glue operators are carried out. The final results are normalized with the momentum and angular momentum sum rules and reported at the physical valence pion mass at \\({\\overline{\\rm {MS}}}\\, (\\mu = 2\\ {\\rm{GeV}})\\). The renormalized momentum fractions for the quarks and glue are \\(\\langle x \\rangle^q = 0.491(20)(23)\\) and \\(\\langle x \\rangle^g = 0.509(20)(23)\\), respectively, and the renormalized total angular momentum fractions for quarks and glue are \\(2 J^q = 0.539(22)(44)\\) and \\(2 J^g = 0.461(22)(44)\\), respectively. The quark spin fraction is \\(\\Sigma = 0.405(25)(37)\\) from our previous work and the quark orbital angular momentum fraction is deduced from \\(2 L^q = 2 J^q - \\Sigma\\) to be \\(0.134(22)(44)\\).

We present the vector, scalar and tensor renormalization constants (RCs) using overlap fermions with either regularization independent momentum subtraction (RI/MOM) or symmetric momentum subtraction (RI/SMOM) as the intermediate scheme on the lattice with lattice spacings \\(a\\) from 0.04 fm to 0.12 fm. Our gauge field configurations from the MILC and RBC/UKQCD collaborations include sea quarks using either the domain wall or the HISQ action, respectively. The results show that RI/MOM and RI/SMOM can provide consistent renormalization constants to the \\(\\overline{\\textrm{MS}}\\) scheme, after proper \\(a^2p^2\\) extrapolations. But at \\(p\\sim 2\\)\\,GeV, both RI/MOM and RI/SMOM suffer from nonperturbative effects which cannot be removed by the perturbative matching. The comparison between the results with different sea actions also suggests that the renormalization constant is discernibly sensitive to the lattice spacing but not to the bare gauge coupling in the gauge action.

We calculate the nucleon isovector scalar charge in lattice QCD using overlap fermions on five ensembles of gauge configurations generated by the RBC/UKQCD collaboration using the domain-wall quark action with \\(2+1\\) dynamical flavors. The five ensembles cover five pion masses, \\(m_\\pi \\approx\\) 139, 171, 302, 337 and 371 MeV, and four lattice spacings, \\(a \\approx \\) 0.06, 0.08, 0.11 and 0.14 fm. Three to six valence quark masses are computed on each ensemble to investigate the pion mass dependence. The extrapolation to the physical pion mass, continuum and infinite volume limits is obtained by a global fit of all data to a formula originated from partially quenched chiral perturbation theory. The excited-states contamination is carefully analyzed with 3--5 sink-source separations and multi-state fits. Our final result, in the \\(\\overline{\\text{MS}}\\) scheme at 2 GeV, is \\(g_{S}^{u-d}= 0.94 (10)_{stat}(8)_{sys}\\), where the first error is the statistical error and the second is the systematic error.

We compute the overlap Dirac spectrum on three gauge ensembles generated using \\(2+1\\)-flavor domain wall fermions. The three ensembles have different lattice spacings and two of them have quark masses tuned to the physical point. The spectral density is determined up to \\(\\lambda\\sim\\)100 MeV with subpercentage statistical uncertainty. We find that the density is close to a constant below \\(\\lambda\\sim\\) 20 MeV as predicted by chiral perturbative theory (\\(\\chi\\)PT), and then increases linearly due to the strange quark mass. By fitting to the next-to-leading order \\(\\chi\\)PT form and using the non-perturbative RI/MOM renormalization, the \\(\\rm SU(2)\\) (keeping the strange quark mass at the physical point) and \\(\\rm SU(3)\\) chiral condensates at \\(\\overline{\\textrm{MS}}\\) 2 GeV are determined to be \\(\\Sigma=(265.4(0.5)(4.2)\\ \\textrm{MeV})^3\\) and \\(\\Sigma_0=(234.3(0.5)(25.8)\\ \\textrm{MeV})^3\\), respectively. The pion decay constants are also determined to be \\(F=84.1(1.9)(8.0)\\) and \\(F_0=58.6(0.5)(10.0)\\) MeV. The systematic errors are carefully estimated including the effects of fitting ranges and the uncertainty of low-energy constant \\(L_6\\). We also show that one can resolve the sea flavor content of the sea quarks and constrain their masses with {\\(\\sim10\\%-20\\%\\)} statistical uncertainties using the Dirac spectral density.

We present a precise calculation of the pion form factor using overlap fermions on seven ensembles of 2+1-flavor domain-wall configurations with pion masses varying from 139 to 340 MeV. Taking advantage of the fast Fourier transform and other techniques to access many combinations of source and sink momenta, we find the pion mean square charge radius to be \\(\\langle {r_\\pi^2} \\rangle= 0.430(5)(13)\\ {\\rm{fm^2}}\\), which agrees well with the experimental result, and includes the systematic uncertainties from chiral extrapolation, lattice spacing and finite-volume dependence. We also find that \\(\\langle {r_\\pi^2} \\rangle\\) depends on both the valence and sea quark masses strongly and predict the pion form factor up to \\(Q^2 = 1.0 \\ {\\rm{GeV^2}}\\) which agrees with experiments very well.

The hadron mass can be obtained through the calculation of the trace of the energy momentum tensor (EMT) in the hadron which includes the trace anomaly and sigma terms. The anomaly due to the conformal symmetry breaking is believed to be an important ingredient for hadron mass generation and confinement. In this work, we will present the calculation of the glue part of the trace anomaly form factors of the pion up to \\(Q^2\\sim 4.3~\\mathrm{GeV}^2\\) and the nucleon up to \\(Q^2\\sim 1~\\mathrm{GeV}^2\\). The calculations are performed on a domain wall fermion (DWF) ensemble with overlap valence quarks at 7 valence pion masses varying from \\(\\sim 250~\\mathrm{MeV}\\) to \\(\\sim 540~\\mathrm{MeV}\\), including the unitary point \\(\\sim 340\\) MeV. We calculate the radius of the glue trace anomaly for the pion and the nucleon from the \\(z\\)-expansion. By performing a two-dimensional Fourier transform on the glue trace anomaly form factors in the infinite momentum frame with no energy transfer, we also obtain their spatial distributions for several valence quark masses. The results are extrapolated to the physical pion mass. We find the pion's form factor changes sign, as does its spatial distribution, for light quark masses. This explains how the trace anomaly contribution to the pion mass approaches zero toward the chiral limit.

We present the first calculation of the hadronic tensor on the lattice for the nucleon. The hadronic tensor can be used to extract the structure functions in deep inelastic scatterings and also provide information for the neutrino-nucleon scattering which is crucial to the neutrino-nucleus scattering experiments at low energies. The most challenging part in the calculation is to solve an inverse problem. We have implemented and tested three algorithms using mock data, showing that the Bayesian Reconstruction method has the best resolution in extracting peak structures while the Backus-Gilbert and Maximum Entropy methods are somewhat more stable for the flat spectral function. Numerical results are presented for both the elastic case (clover fermions on domain wall configuration with \\(m_\\pi\\sim\\) 370 MeV and \\(a\\sim\\) 0.06 fm) and a case (anisotropic clover lattice with \\(m_\\pi\\sim\\) 380 MeV and \\(a_t\\sim\\) 0.035 fm) with large momentum transfer. For the former case, the reconstructed Minkowski hadronic tensor gives precisely the vector charge which proves the feasibility of the approach. While for the latter case, the nucleon resonances and possibly shallow inelastic scattering contributions around \\(\\nu=1\\) GeV are clearly observed but no information is obtained for higher excited states with \\(\\nu>2\\) GeV. A check of the effective masses of \\(\\rho\\) meson with different lattice setups indicates that, in order to reach higher energy transfers, using lattices with smaller lattice spacings is essential.